Dynamics of bubbles under stochastic pressure forcing

Several studies have investigated the dynamics of a single spherical bubble at rest under a nonstationary pressure forcing. However, attention has almost always been focused on periodic pressure oscillations, neglecting the case of stochastic forcing. This fact is quite surprising, as random pressure fluctuations are widespread in many applications involving bubbles (e.g., hydrodynamic cavitation in turbulent flows or bubble dynamics in acoustic cavitation), and noise, in general, is known to induce a variety of counterintuitive phenomena in nonlinear dynamical systems such as bubble oscillators. To shed light on this unexplored topic, here we study bubble dynamics as described by the Keller-Miksis equation, under a pressure forcing described by a Gaussian colored noise modeled as an Ornstein-Uhlenbeck process. Results indicate that, depending on noise intensity, bubbles display two peculiar behaviors: when intensity is low, the fluctuating pressure forcing mainly excites the free oscillations of the bubble, and the bubble's radius undergoes small amplitude oscillations with a rather regular periodicity. Differently, high noise intensity induces chaotic bubble dynamics, whereby nonlinear effects are exacerbated and the bubble behaves as an amplifier of the external random forcing.

Figure 1

(a) Example of a time-series of the normalized stochastic pressure forcing $p\left(t\right)/\overline{p}$. (b) Time-series of the normalized radius $R\left(t\right)/R_{\text{eq}}$ of the bubble forced by the pressure reported in (a). The black dots in (b) highlight the bubble radius attained at the instants $n{T}_n$, where $n$ is an integer and $T_n$ is the natural oscillation period of the bubble (see Sec. 3a for the explanation). (c),(d) Probability density functions, and (e),(f) autocorrelation functions of the time-series (partially) reported in (a) and (b). The dashed lines in (e) and (f) mark the level ${\rho }_p={\rho }_R=0.1$. It should be noted that the time-lag ${\widehat{t}}_l$ such that $\rho \left({\widehat{t}}_l\right)=0.1$ is defined as the correlation time of the time-series. The times in (a) and (b) and the time lags reported in (e) and (f) are normalized by $T_n$. The adopted parameters are ${\sigma }_p=60\times {10}^3$ Pa and ${\tau }_p=2.0T_n=2.8\times {10}^{-6}$ s.

Figure 2

(a),(b) Noise-intensity bifurcation diagrams. For a given value of ${\sigma }_p/\overline{p}$, the dynamics of $R\left(t\right)$ is simulated for $4000T_n$. From this simulation, only the values $R\left(n{T}_n\right)/R_{\text{eq}}$ are selected, and they are reported in the vertical axis for the given ${\sigma }_p/\overline{p}$. In both panels, the gray circles refer to ${\tau }_p=T_n$. In panels (a) and (b), the red dots refer to ${\tau }_p=T_n/2$ and ${\tau }_p=2T_n$, respectively. (c)–(f) Time segments of the time-series $R\left(t\right)/R_{\text{eq}}$ and $p\left(t\right)/\overline{p}$. The horizontal dotted lines mark the equilibrium radius $R\left(t\right)/R_{\text{eq}}=1$ and the mean pressure $p\left(t\right)/\overline{p}=1$. The black dots in (c) and (d) highlight the bubble radius attained at the instants $n{T}_n$. Panels (c) and (e) refer to ${\sigma }_p/\overline{p}=0.3$, and panels (d) and (f) refer to ${\sigma }_p/\overline{p}=0.4$; in both cases, ${\tau }_p=T_n$.

Figure 3

Time segments of the time-series $R\left(t\right)/R_{\text{eq}}$ (top row) and $p\left(t\right)/\overline{p}$ (bottom row) when a sinusoidal external pressure—with amplitude $A_p=\sqrt{2}{\sigma }_p$ and period ${\tau }_p=T_n$—is applied. The dotted lines mark the equilibrium radius $R\left(t\right)/R_{\text{eq}}=1$ and the mean pressure $p\left(t\right)/\overline{p}=1$. The black dots in (a)–(c) highlight the bubble radius attained at the instants $n{T}_n$. Panels (a),(d), (b),(e), and (c),(f) refer to ${\sigma }_p/\overline{p}=0.4, {\sigma }_p/\overline{p}=1.2$, and ${\sigma }_p/\overline{p}=2.0$, respectively.

Figure 4

Effect of ${\sigma }_p/\overline{p}$ on some relevant statistical parameters that describe the time-series $R\left(t\right)$.

Figure 5

Effect of the coefficient of variation of the pressure, $c_{V,p}$, on the coefficient of variation of the bubble radius, $c_{V,R}$. The shaded zone highlights the lower half-plane bounded by the bisector, where the bubble exhibits “damper” behavior. In the upper half-plane, the bubble behaves as an “amplifier.”

Figure 6

(a) Effect of the noise intensity ${\sigma }_p/\overline{p}$ on the ratio between the integral scale of the radius time-series, $I_R$, and the integral scale of the pressure forcing, $I_p$. The gray zone highlights the condition $I_R<I_p$. Autocorrelation diagrams of $R\left(t\right)$ [panels (A1) and (B1)] and $p\left(t\right)$ [panels (A2) and (B2)]. The red lines mark the level where the autocorrelation function is 0.1. (A3),(B3) Power (amplitude) spectrum of $R\left(t\right)$. It should be noted that the horizontal axis reports the period of the $k\mathrm{th}$ harmonics (rather than its frequency). (A4)–(B5) Relevant time segment of the time-series $R\left(t\right)$ and $p\left(t\right)$. The dotted lines mark the equilibrium radius $R_{\text{eq}}$ and the mean pressure $\overline{p}$.

Figure 7

Time-series of $R\left(t\right)/R_{\text{eq}}$ in four relevant time segments in the case of ${\tau }_p=T_n/2$ (a) and ${\tau }_p=2T_n$ (b). In both cases, ${\sigma }_p/\overline{p}=0.70$. The red dots plotted at $R\left(t\right)/R_{\text{eq}}=1$ mark the instants when $p\left(t\right)<\overline{p}$, and they should not be confused with the dynamics of $R\left(t\right)/R_{\text{eq}}$ reported by the black line. Panels (a) and (b) report different ranges in the vertical axis.

Figure 8

(a,b) Probability density function of the metric $R/R_{\text{eq}}$. (c) Complementary cumulative distribution function of $R/R_{\text{eq}}$ evaluated for $R/R_{\text{eq}}>1$ (right tail of the distribution). (d) Cumulative distribution function of $R/R_{\text{eq}}$ evaluated for $R/R_{\text{eq}}<1$ (left tail of the distribution); note that the horizontal axis reports $R_{\text{eq}}/R$ and not $R/R_{\text{eq}}$ as in panel (c).

Figure 9

(a) Example of curves $R(\mathrm{\Delta }t,t)$ numerically computed adopting different time steps $\mathrm{\Delta }t$. (b) Relative error ${\varepsilon }_R\left(t\right)$ occurring in the numerical computation performed with different time steps. The relative error is evaluated considering the curve computed with $\mathrm{\Delta }t={10}^{-9}$ s as the exact reference. The initial conditions are $R\left(0\right)/R_{\text{eq}}=2$ and $\dot{R}\left(0\right)=0$. The pressure field is uniform.

Figure 10

Effect of the duration $T$ of the simulation on the statistical metrics that describes the time-series $R\left(t\right)$. Similar to Fig. 4, two statistical parameters and their dependence on ${\sigma }_p/\overline{p}$ are considered. The different curves were evaluated considering different lengths of the simulation. The parameter ${\tau }_p=T_n$ is adopted.

Figure 11

Effect of different realizations of the stochastic process on the statistical parameters that describe the time-series $R\left(t\right)$. Similar to Fig. 4, two statistical parameters and their dependence on ${\sigma }_p/\overline{p}$ are considered. The different curves were evaluated with the same noise intensities and correlation times, but with a different set of random numbers. The parameter ${\tau }_p=T_n$ is adopted.